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PROFESSOR: The final quiz is
scheduled for a week from
6
00:00:20,350 --> 00:00:23,330
today, on December 8.
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00:00:23,330 --> 00:00:28,230
Following that weekend, we'll
have one class on the 13th.
8
00:00:28,230 --> 00:00:32,950
And the last days of classes
are Monday, Tuesday, and
9
00:00:32,950 --> 00:00:36,110
Wednesday, the 12th,
13th, and 14th.
10
00:00:36,110 --> 00:00:39,030
So I know a number of you spoke
up last time and said
11
00:00:39,030 --> 00:00:42,840
that you have a conflict on
Thursday, the eighth, with a
12
00:00:42,840 --> 00:00:44,250
quiz in another class.
13
00:00:44,250 --> 00:00:46,910
So could I see a show of hands
again of how many of you have
14
00:00:46,910 --> 00:00:49,080
this conflict?
15
00:00:49,080 --> 00:00:55,260
So three, six people, seven
people, you're going to enroll
16
00:00:55,260 --> 00:00:56,475
in the other course just
so you can put
17
00:00:56,475 --> 00:00:57,725
off taking the quiz.
18
00:01:02,440 --> 00:01:04,730
That's unfortunate, I don't
think anybody should have to
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00:01:04,730 --> 00:01:06,590
take two quizzes in one day.
20
00:01:06,590 --> 00:01:07,940
We can't move it up.
21
00:01:07,940 --> 00:01:09,430
We'll have to move it back.
22
00:01:09,430 --> 00:01:14,670
So I don't know if I am
violating any Institute rule,
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00:01:14,670 --> 00:01:19,200
but I know that it is strictly
illegal to give assignments
24
00:01:19,200 --> 00:01:22,090
that are due after the last day
of classes, let alone have
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00:01:22,090 --> 00:01:25,180
a quiz after the last day of
classes, which has not been
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00:01:25,180 --> 00:01:28,000
previously scheduled as
a final examination.
27
00:01:28,000 --> 00:01:31,560
So we can't give it after
Wednesday, the 14th, the last
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00:01:31,560 --> 00:01:32,680
day of classes.
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00:01:32,680 --> 00:01:37,700
So, do any of the six impacted
people have a strong
30
00:01:37,700 --> 00:01:41,975
preference for when we should
schedule the quiz?
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00:01:41,975 --> 00:01:44,200
AUDIENCE: Monday?
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00:01:44,200 --> 00:01:45,790
PROFESSOR: Due it Monday?
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00:01:45,790 --> 00:01:48,050
It doesn't have to
be that soon.
34
00:01:48,050 --> 00:01:48,470
Monday?
35
00:01:48,470 --> 00:01:50,860
AUDIENCE: Why not Tuesday?
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00:01:50,860 --> 00:01:52,190
PROFESSOR: Because Tuesday
we have a class.
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00:01:52,190 --> 00:01:53,480
AUDIENCE: Or Wednesday?
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00:01:53,480 --> 00:01:55,352
Well we would have a Thursday
class instead.
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00:01:55,352 --> 00:01:57,220
Correct?
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00:01:57,220 --> 00:01:59,610
PROFESSOR: No.
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00:01:59,610 --> 00:02:01,940
If we're doing it after
the quiz next
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00:02:01,940 --> 00:02:05,100
week, there's a class.
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00:02:05,100 --> 00:02:07,310
We have a last meeting
here on the 13th.
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00:02:07,310 --> 00:02:10,210
And I'm not going to shift
it for other people,
45
00:02:10,210 --> 00:02:11,460
just for the impacted.
46
00:02:16,920 --> 00:02:20,260
So of the six who are entitled
to vote, how many would prefer
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00:02:20,260 --> 00:02:25,320
to have it Monday, the 12th?
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00:02:25,320 --> 00:02:28,230
Three and I know how it's
going to turn out.
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00:02:28,230 --> 00:02:31,570
And for Wednesday, the 14th?
50
00:02:31,570 --> 00:02:32,820
Three.
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00:02:36,412 --> 00:02:39,014
AUDIENCE: I can't take the test
because-- not because of
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00:02:39,014 --> 00:02:39,828
another test--
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00:02:39,828 --> 00:02:40,316
I'm out of town.
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00:02:40,316 --> 00:02:41,780
And I might come
back Saturday.
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00:02:41,780 --> 00:02:45,684
And Monday is just a little
quick to take it
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00:02:45,684 --> 00:02:46,934
after we get back.
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00:02:51,245 --> 00:02:54,020
PROFESSOR: I think I'm going to
have to make an executive--
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00:02:54,020 --> 00:02:54,841
yes
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00:02:54,841 --> 00:02:57,146
AUDIENCE: Professor, when you
mean by conflict, do you mean
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00:02:57,146 --> 00:02:59,090
having an exam at
the exact time?
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00:02:59,090 --> 00:03:02,690
PROFESSOR: No, it's on
the exact same day.
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00:03:02,690 --> 00:03:05,760
Considering this is a two hour
examination, to have another
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00:03:05,760 --> 00:03:09,620
examination exactly the same
day, if you're not brain dead
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00:03:09,620 --> 00:03:13,420
after an hour or two, you will
be pretty close to it.
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00:03:13,420 --> 00:03:21,840
So I think that is an unfair
penalty to pay.
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00:03:21,840 --> 00:03:27,230
I think, to keep it as close as
possible to the quiz that
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00:03:27,230 --> 00:03:29,330
the rest of the people will be
taking, why don't we make it--
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00:03:29,330 --> 00:03:32,170
since it's a tie vote--
on Monday, the 12th.
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00:03:32,170 --> 00:03:34,380
Do it sooner, rather
than later.
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00:03:34,380 --> 00:03:38,000
And I'll let you know next time
of where we will hold it.
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00:03:38,000 --> 00:03:40,930
I'll have to arrange
a room for it.
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00:03:40,930 --> 00:03:45,480
So that will not be
a make-up quiz.
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00:03:45,480 --> 00:03:47,220
It would be a made-up--
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00:03:47,220 --> 00:03:47,980
invented--
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00:03:47,980 --> 00:03:49,230
quiz.
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00:03:53,530 --> 00:03:57,150
And, as I say on Monday, I
should have all of the
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00:03:57,150 --> 00:04:01,960
homework and the previous
quiz to return to you.
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00:04:01,960 --> 00:04:04,620
Reason you don't have it now
is all the little crabbed
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00:04:04,620 --> 00:04:07,450
handwriting that you see before
you in the form of
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00:04:07,450 --> 00:04:11,470
these notes, which
takes forever.
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00:04:11,470 --> 00:04:15,120
All right, so since we've
satisfied the unpleasant
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00:04:15,120 --> 00:04:18,839
aspects of the end of the
term, let's get back to
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00:04:18,839 --> 00:04:22,390
discussing some of the other
piezoelectric effects that we
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00:04:22,390 --> 00:04:23,500
have defined.
85
00:04:23,500 --> 00:04:27,850
And then also ask the question
rhetorically, is there
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00:04:27,850 --> 00:04:33,660
anything like a representation
surface for a third-ranked
87
00:04:33,660 --> 00:04:35,290
tensor property?
88
00:04:35,290 --> 00:04:37,110
And it doesn't look promising.
89
00:04:37,110 --> 00:04:40,500
But there are some things that
we can do to discuss variation
90
00:04:40,500 --> 00:04:42,590
of properties with direction,
and we'll see
91
00:04:42,590 --> 00:04:43,980
directly what those are.
92
00:04:43,980 --> 00:04:47,230
But first let's look at the
converse piezoelectric effect.
93
00:04:55,540 --> 00:04:59,860
And this again is a third-ranked
tensor property,
94
00:04:59,860 --> 00:05:05,077
but what we do is to have the
elements of strain, epsilon
95
00:05:05,077 --> 00:05:09,350
ij, a second-ranked tensor.
96
00:05:09,350 --> 00:05:15,190
And to take advantage of this
curious relation between the
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00:05:15,190 --> 00:05:18,540
directed converse effects,
we define the converse
98
00:05:18,540 --> 00:05:23,330
piezoelectric effect as giving
you nine elements of strain in
99
00:05:23,330 --> 00:05:29,210
terms of a third-ranked tensor
dijk times e sub i.
100
00:05:29,210 --> 00:05:33,660
So what is not standard is our
convention for the order of
101
00:05:33,660 --> 00:05:36,260
the subscripts on the moduli.
102
00:05:36,260 --> 00:05:39,420
And we make up the rules, we
can do it any way we like.
103
00:05:39,420 --> 00:05:42,640
And the advantage of defining
it this way, in nonstandard
104
00:05:42,640 --> 00:05:46,710
tensor notation, is that we can
use the same coefficients
105
00:05:46,710 --> 00:05:50,616
for both the direct and
the converse effects.
106
00:05:50,616 --> 00:05:54,030
So let me-- to illustrate what
these equations look like--
107
00:05:54,030 --> 00:05:57,270
write out a few examples.
108
00:05:57,270 --> 00:06:07,026
Epsilon 11 would be d1jk
times e sub 1.
109
00:06:07,026 --> 00:06:09,980
I'm writing it in simple
fashion, and not expanding
110
00:06:09,980 --> 00:06:12,830
fully, just to save
space and time.
111
00:06:12,830 --> 00:06:24,820
Epsilon 22 would be
d2jk times e2.
112
00:06:24,820 --> 00:06:33,760
Epsilon 33 will be d3jk
times epsilon 3.
113
00:06:33,760 --> 00:06:37,210
And let me stop after these six
terms, and write, at least
114
00:06:37,210 --> 00:06:38,770
for these, an expansion.
115
00:06:38,770 --> 00:06:42,040
Because this gives us some
interesting information.
116
00:06:44,750 --> 00:06:48,820
I'm not doing the equal
signs in here.
117
00:06:48,820 --> 00:06:52,920
So this would say that the
element of strain epsilon 11
118
00:06:52,920 --> 00:07:08,990
is d111 times e1 plus d112 times
e2 plus d113 times e3.
119
00:07:08,990 --> 00:07:11,270
And I want to say this is 11k.
120
00:07:11,270 --> 00:07:13,180
And I want to say that
this is 22k.
121
00:07:20,280 --> 00:07:26,610
The next term would be epsilon
22, and that would be d122
122
00:07:26,610 --> 00:07:35,850
times epsilon 2 plus d212.
123
00:07:35,850 --> 00:07:37,802
AUDIENCE: Wouldn't the
2's be balanced?
124
00:07:37,802 --> 00:07:38,290
PROFESSOR: Hm?
125
00:07:38,290 --> 00:07:39,760
AUDIENCE: Wouldn't
that be d2's?
126
00:07:39,760 --> 00:07:40,320
PROFESSOR: Yeah, you're right.
127
00:07:40,320 --> 00:07:52,530
This should be epsilon jk equals
dijk times e sub i.
128
00:07:52,530 --> 00:07:54,270
So this is all e1.
129
00:07:54,270 --> 00:07:55,940
This is e1.
130
00:07:55,940 --> 00:07:57,760
And this one goes
with this one.
131
00:07:57,760 --> 00:08:03,230
This 2 goes with this one.
132
00:08:03,230 --> 00:08:04,580
You're right.
133
00:08:04,580 --> 00:08:10,710
d2 and this is the
d311 times e11.
134
00:08:10,710 --> 00:08:18,520
This would be 122 times
e1 plus 222 times e2
135
00:08:18,520 --> 00:08:26,830
plus d322 times e3.
136
00:08:26,830 --> 00:08:33,590
And the fourth one would be e33
equals d133 times e1 plus
137
00:08:33,590 --> 00:08:43,679
d233 times e2 plus
d333 times e3.
138
00:08:46,490 --> 00:08:52,760
OK, these elements here are the
three that appear in the
139
00:08:52,760 --> 00:08:57,930
box that I had indicated
for the terms djk.
140
00:09:03,530 --> 00:09:07,410
When we write the direct
piezoelectric effect, this
141
00:09:07,410 --> 00:09:09,190
would be the box of
coefficients.
142
00:09:09,190 --> 00:09:12,290
When we write the converse
effect, this is the box of
143
00:09:12,290 --> 00:09:13,650
coefficients.
144
00:09:13,650 --> 00:09:18,550
And now what I wanted to point
out is that delta v over v is
145
00:09:18,550 --> 00:09:23,610
equal to episilon 11 plus
epsilon 22 plus epsilon 33,
146
00:09:23,610 --> 00:09:26,210
which is the trace of
the strain tensor.
147
00:09:26,210 --> 00:09:29,400
And we're going to get one set
of terms which depends on the
148
00:09:29,400 --> 00:09:33,290
x1 component of the field,
another set of three terms
149
00:09:33,290 --> 00:09:37,025
that depend on the x2 component
of the field, and
150
00:09:37,025 --> 00:09:38,650
another one on the
x3 component.
151
00:09:43,230 --> 00:09:46,230
If these expressions sum
to 0, then there would
152
00:09:46,230 --> 00:09:47,770
be no volume change.
153
00:09:47,770 --> 00:09:51,320
So this first set of 3--
154
00:09:51,320 --> 00:09:53,070
3 of the first 6 equations--
155
00:09:53,070 --> 00:09:56,270
give you an indication of when
the application of a field
156
00:09:56,270 --> 00:09:59,270
will result in no volume
change in the sample.
157
00:09:59,270 --> 00:10:03,300
And that again, I remind you,
can be for two reasons.
158
00:10:03,300 --> 00:10:10,180
It could be because all of
these six of the nine
159
00:10:10,180 --> 00:10:12,430
piezoelectric moduli are 0.
160
00:10:12,430 --> 00:10:17,720
Or alternatively, if you
examined the form of the
161
00:10:17,720 --> 00:10:21,750
piezoelectric moduli matrices
that is required by symmetry
162
00:10:21,750 --> 00:10:25,880
constraints, you'll find that
there are a substantial number
163
00:10:25,880 --> 00:10:29,750
of point groups for which
some terms are 0.
164
00:10:29,750 --> 00:10:32,870
But then there is, in addition,
an equality between
165
00:10:32,870 --> 00:10:36,840
some of the other terms, which
make the volume change zero.
166
00:10:36,840 --> 00:10:42,250
Even though, not all of the
elements within this 1/2 box
167
00:10:42,250 --> 00:10:44,270
of moduli are zero.
168
00:10:44,270 --> 00:10:47,410
So that is an interesting
effect.
169
00:10:47,410 --> 00:10:49,970
And it turns out that
the majority of the
170
00:10:49,970 --> 00:10:52,580
non-centrosymmetric point groups
do not have a volume
171
00:10:52,580 --> 00:10:56,730
change, when you apply in a
field in any way you choose.
172
00:10:56,730 --> 00:11:01,720
A few do, but it's a minority.
173
00:11:01,720 --> 00:11:03,920
Alright, but this is not the
main point of writing this.
174
00:11:03,920 --> 00:11:05,900
I want to--
175
00:11:05,900 --> 00:11:07,170
at this point--
176
00:11:07,170 --> 00:11:10,190
point out that this tells you
about the volume change.
177
00:11:10,190 --> 00:11:14,330
And then we would have
additional terms, we would
178
00:11:14,330 --> 00:11:33,740
have a term of the form e1
something like e123 or e132.
179
00:11:33,740 --> 00:11:37,350
And this would be e14.
180
00:11:37,350 --> 00:11:41,710
This would also be e14 since
we replace both of those
181
00:11:41,710 --> 00:11:44,260
subscripts by a single
subscript.
182
00:11:44,260 --> 00:11:46,220
And the tensor elements
that would go in
183
00:11:46,220 --> 00:11:52,430
here would be d123.
184
00:11:52,430 --> 00:11:54,000
We just want one of the them.
185
00:11:54,000 --> 00:12:09,300
d123 times e1, and then we want
d223 times e2, and then
186
00:12:09,300 --> 00:12:14,520
d323 times e3.
187
00:12:14,520 --> 00:12:21,940
We're writing one of the
specific equations for the
188
00:12:21,940 --> 00:12:23,500
shear strings.
189
00:12:23,500 --> 00:12:29,600
If we write the expression for
d132, this is going to be d132
190
00:12:29,600 --> 00:12:42,400
times e1 plus d232 times
e2 plus d332 times e3.
191
00:12:46,060 --> 00:12:49,690
There are two interesting
consequences of this.
192
00:12:49,690 --> 00:12:55,060
The strain tensor
is symmetric.
193
00:12:55,060 --> 00:13:00,130
And this element of strain--
194
00:13:00,130 --> 00:13:03,360
why have I got three
subscripts in here?
195
00:13:03,360 --> 00:13:04,840
Don't want that one in there.
196
00:13:07,850 --> 00:13:09,570
These two strains are equal.
197
00:13:09,570 --> 00:13:15,710
And therefore, if we would apply
just an e1 for example,
198
00:13:15,710 --> 00:13:21,980
let e be equal to just a
component of field along x1.
199
00:13:21,980 --> 00:13:24,760
The strains have to be equal.
200
00:13:24,760 --> 00:13:29,690
But we have two different
tensor elements here.
201
00:13:29,690 --> 00:13:32,460
And the only way that strain
can be symmetric, and it's
202
00:13:32,460 --> 00:13:41,460
defined as such, is that d123
be identical to d132.
203
00:13:41,460 --> 00:13:44,220
And that resolves the issue
that came up in connection
204
00:13:44,220 --> 00:13:46,080
with the direct case
electric effect.
205
00:13:46,080 --> 00:13:48,480
We said the direct piezoelectric
effect depends
206
00:13:48,480 --> 00:13:53,850
just on the sum of the elements
dijk and dikj.
207
00:13:53,850 --> 00:13:56,610
And since they're lumped
together, all we can measure
208
00:13:56,610 --> 00:13:58,740
is the sum.
209
00:13:58,740 --> 00:14:03,160
And, so, we'll just have
to call that a
210
00:14:03,160 --> 00:14:05,080
single matrix element.
211
00:14:05,080 --> 00:14:08,540
The converse piezoelectric
effect tells us these tensor
212
00:14:08,540 --> 00:14:12,200
elements have to be equal,
if the strain
213
00:14:12,200 --> 00:14:13,450
tensor is to be symmetric.
214
00:14:16,350 --> 00:14:26,260
So that says that, since we
defined d16 as the sum of d123
215
00:14:26,260 --> 00:14:39,650
plus d132, this says that d132
is equal to d123 is equal to
216
00:14:39,650 --> 00:14:47,030
1/2 of d16, just making
the equality
217
00:14:47,030 --> 00:14:49,020
in the reverse direction.
218
00:14:49,020 --> 00:14:55,860
The converse effect let's us
say that 123 has to be 132,
219
00:14:55,860 --> 00:15:01,210
that any ijk has to be
equal to a dijk.
220
00:15:01,210 --> 00:15:05,880
So if I tried now to write this
first expression in the
221
00:15:05,880 --> 00:15:20,230
reduced subscript notation,
e23 is what we let e5 be.
222
00:15:20,230 --> 00:15:26,510
And now we have this equal to
and in our reduced subscripts,
223
00:15:26,510 --> 00:15:31,370
we have this as 1/2 of d16.
224
00:15:31,370 --> 00:15:34,150
And the field that's
multiplying this is
225
00:15:34,150 --> 00:15:36,480
piezoelectric modulus is e1.
226
00:15:40,880 --> 00:15:45,470
And that messy factor of 2 has
come back to haunt us again.
227
00:15:45,470 --> 00:15:46,955
It's like trying to stuff
a jack-in-the
228
00:15:46,955 --> 00:15:48,530
box back in the box.
229
00:15:48,530 --> 00:15:49,810
It keeps popping up.
230
00:15:49,810 --> 00:15:56,550
We ate the factor of 2 in
defining the matrix
231
00:15:56,550 --> 00:15:59,050
representation of the
232
00:15:59,050 --> 00:16:00,280
piezoelectric electric modulus.
233
00:16:00,280 --> 00:16:03,920
And now when we try to go to a
reduced subscript notation for
234
00:16:03,920 --> 00:16:06,210
the converse piezoelectric
effect, we've
235
00:16:06,210 --> 00:16:07,780
got a 1/2 in there.
236
00:16:07,780 --> 00:16:12,370
And similarly, the second
equation would be e5--
237
00:16:12,370 --> 00:16:14,670
same result epsilon 5--
238
00:16:14,670 --> 00:16:22,050
and it's 132, but 132 is
1/2 of d16 times e1.
239
00:16:22,050 --> 00:16:33,000
And we have a similar mess for
the other coefficients here.
240
00:16:33,000 --> 00:16:34,230
So what do we do?
241
00:16:34,230 --> 00:16:46,800
Do we say that the relation
between strain epsilon j
242
00:16:46,800 --> 00:16:54,820
equals dij e sub j has 1/2 in
front of several of the
243
00:16:54,820 --> 00:16:57,330
coefficients and
not in others?
244
00:16:57,330 --> 00:17:00,670
Well, we can't really absorb
the factor of 2 in the
245
00:17:00,670 --> 00:17:04,079
definition of the piezoelectric
moduli, because
246
00:17:04,079 --> 00:17:05,819
we've already done that.
247
00:17:05,819 --> 00:17:11,160
So the only thing we can do is
to say that we will have to
248
00:17:11,160 --> 00:17:16,710
take the off diagonal strains,
and define them as having a
249
00:17:16,710 --> 00:17:19,730
1/2 in front, and
we add these up.
250
00:17:25,640 --> 00:17:30,720
So we will have to write matrix
strain in this reduced
251
00:17:30,720 --> 00:17:32,770
subscript notation.
252
00:17:32,770 --> 00:17:37,560
We'll have to take e11, epsilon
12, epsilon 13,
253
00:17:37,560 --> 00:17:45,250
epsilon 21, epsilon 22, epsilon
23, epsilon 31,
254
00:17:45,250 --> 00:17:50,790
epsilon 32, and epsilon 33.
255
00:17:50,790 --> 00:17:54,490
And in converting this to matrix
form, we'll call this
256
00:17:54,490 --> 00:17:57,110
epsilon 1, analogous
to what we did
257
00:17:57,110 --> 00:17:58,710
for the tensile stresses.
258
00:17:58,710 --> 00:18:02,980
We'll call this epsilon
2 and this epsilon 3.
259
00:18:02,980 --> 00:18:06,960
And then for all of the
off-diagonal elements of
260
00:18:06,960 --> 00:18:11,250
strain, in order to avoid the
factor of 2 popping up in
261
00:18:11,250 --> 00:18:13,890
front of the matrix
representation of the
262
00:18:13,890 --> 00:18:17,280
piezoelectric moduli, we're
going to have to put in here
263
00:18:17,280 --> 00:18:23,867
1/2 of epsilon 4, 1/2 of epsilon
5, and 1/2 of epsilon
264
00:18:23,867 --> 00:18:28,050
6, and same for the off diagonal
terms 1/2 of epsilon
265
00:18:28,050 --> 00:18:33,035
5, 1/2 of epsilon 4, and
1/2 of epsilon 6.
266
00:18:36,290 --> 00:18:40,340
So the moral of this story is
that you can't win, but if you
267
00:18:40,340 --> 00:18:44,410
play it right, you can
come out even.
268
00:18:44,410 --> 00:18:49,060
So only if we define the reduced
subscript strains in
269
00:18:49,060 --> 00:18:52,510
this fashion, can we write an
expression of this form.
270
00:19:05,140 --> 00:19:08,370
So this algebra is carried
through for you for the other
271
00:19:08,370 --> 00:19:10,890
elements in the notes.
272
00:19:10,890 --> 00:19:15,080
But this is the way we are
forced, unless we want to have
273
00:19:15,080 --> 00:19:18,620
a factor of 2 in some terms, and
not in others, is the way
274
00:19:18,620 --> 00:19:20,830
we have to define
matrix strain.
275
00:19:26,700 --> 00:19:30,850
All this is formalism and
definition, but I'd like to
276
00:19:30,850 --> 00:19:32,110
now do two things.
277
00:19:32,110 --> 00:19:35,240
First of all, give you some
examples of real numbers for
278
00:19:35,240 --> 00:19:41,390
piezoelectric moduli, and then
ask the question about
279
00:19:41,390 --> 00:19:42,765
representation surfaces.
280
00:19:45,500 --> 00:19:49,170
Once again, these numbers are in
the handout for you, so you
281
00:19:49,170 --> 00:19:51,010
don't have to make
note of them.
282
00:19:51,010 --> 00:19:58,280
But one of the very important
piezoelectric materials is the
283
00:19:58,280 --> 00:20:00,020
quartz form of SiO2.
284
00:20:07,060 --> 00:20:10,160
SiO2 has many polymorphic
forms.
285
00:20:10,160 --> 00:20:13,820
Quartz is the form that's stable
at room temperature,
286
00:20:13,820 --> 00:20:25,080
and it has point group
32 asymmetric.
287
00:20:25,080 --> 00:20:28,640
And there are higher
temperature
288
00:20:28,640 --> 00:20:30,820
polymorphs of SiO2.
289
00:20:30,820 --> 00:20:34,090
There's a phase transition in
quartz to a more symmetric
290
00:20:34,090 --> 00:20:37,550
form, and then there are cubic
forms at the highest
291
00:20:37,550 --> 00:20:38,800
temperatures.
292
00:20:40,670 --> 00:20:45,970
Now, quartz is not the material
that displays the
293
00:20:45,970 --> 00:20:48,750
largest piezoelectric moduli.
294
00:20:48,750 --> 00:20:50,210
But it has the following
advantages.
295
00:20:50,210 --> 00:20:55,020
One is it is a naturally
occurring material that is
296
00:20:55,020 --> 00:20:56,510
very inexpensive.
297
00:20:56,510 --> 00:20:59,260
So it's not an exotic
expensive material.
298
00:20:59,260 --> 00:21:00,260
Very stable.
299
00:21:00,260 --> 00:21:02,360
It's not water soluble.
300
00:21:02,360 --> 00:21:03,620
Extremely tough.
301
00:21:03,620 --> 00:21:06,620
You can take a thin wafer of
quartz, for example, if you
302
00:21:06,620 --> 00:21:10,550
want to make a monochromator for
a diffraction experiment.
303
00:21:10,550 --> 00:21:13,550
You can take a thin wafer of
quartz, and bend it like this,
304
00:21:13,550 --> 00:21:17,910
and it does not break,
very elastic.
305
00:21:17,910 --> 00:21:22,240
And if you want a material
that's going to earn its
306
00:21:22,240 --> 00:21:26,130
living by being squished, you
want something that doesn't
307
00:21:26,130 --> 00:21:29,480
plastically deform and something
that is very hard
308
00:21:29,480 --> 00:21:31,300
and resistant to stress.
309
00:21:31,300 --> 00:21:35,070
So quartz, even though the
moduli are not the largest, is
310
00:21:35,070 --> 00:21:37,840
a very attractive material,
and is used in
311
00:21:37,840 --> 00:21:40,730
a variety of devices.
312
00:21:40,730 --> 00:21:43,760
There was a time when CB radios
313
00:21:43,760 --> 00:21:45,260
were very, very popular.
314
00:21:45,260 --> 00:21:48,290
Everybody had to have
one in their car.
315
00:21:48,290 --> 00:21:51,990
I guess so they could pretend
that they were truck drivers.
316
00:21:51,990 --> 00:21:54,550
But anyway, you don't have
them anymore now that
317
00:21:54,550 --> 00:21:57,690
cellphones have come
in in existence.
318
00:21:57,690 --> 00:22:01,310
But for your CB radio,
you needed
319
00:22:01,310 --> 00:22:04,150
something called a crystal.
320
00:22:04,150 --> 00:22:07,000
And they were fairly
expensive.
321
00:22:07,000 --> 00:22:10,020
And the number of channels on
which you could communicate
322
00:22:10,020 --> 00:22:12,870
dependent on the number of
crystals that you could plug
323
00:22:12,870 --> 00:22:15,090
into your CB radio.
324
00:22:15,090 --> 00:22:17,820
The so-called crystal
was exactly that.
325
00:22:17,820 --> 00:22:20,550
It was a little black box that
looked almost like a
326
00:22:20,550 --> 00:22:21,360
transistor.
327
00:22:21,360 --> 00:22:23,180
And there were two leads
coming out of it.
328
00:22:23,180 --> 00:22:26,300
If you ever got curious and
broke this thing open, what
329
00:22:26,300 --> 00:22:30,120
you found was a nice
wafer of quartz.
330
00:22:30,120 --> 00:22:32,500
And on the wafer of quartz--
brazed onto it--
331
00:22:32,500 --> 00:22:33,470
were two wires.
332
00:22:33,470 --> 00:22:35,040
And that's all there
was in the box.
333
00:22:35,040 --> 00:22:37,240
The crystal really
was a crystal.
334
00:22:37,240 --> 00:22:40,580
And the crystals had been
very precisely ground to
335
00:22:40,580 --> 00:22:45,720
thicknesses such that when a
field caused these wafers to
336
00:22:45,720 --> 00:22:51,210
hit a resonance, that resonance
would be at exactly
337
00:22:51,210 --> 00:22:52,430
a particular frequency.
338
00:22:52,430 --> 00:22:54,170
And that was the frequency
of that channel.
339
00:22:57,350 --> 00:23:03,610
One of the crises, during the
Second World War, is that the
340
00:23:03,610 --> 00:23:08,070
highest quality natural crystals
of quartz come from
341
00:23:08,070 --> 00:23:15,220
Brazil, and during the conflict
the sea channels were
342
00:23:15,220 --> 00:23:16,380
essentially blocked.
343
00:23:16,380 --> 00:23:19,650
And so, people-- in order
to make all these
344
00:23:19,650 --> 00:23:21,240
communication devices--
345
00:23:21,240 --> 00:23:25,370
had to learn how to synthesize
crystals of quartz
346
00:23:25,370 --> 00:23:26,790
synthetically.
347
00:23:26,790 --> 00:23:29,420
And there are a number of
companies, such as Sylvania up
348
00:23:29,420 --> 00:23:32,210
on the North Shore, that
developed entire buildings
349
00:23:32,210 --> 00:23:34,680
devoted to growing single
crystals of quartz.
350
00:23:34,680 --> 00:23:38,040
And they're big tanks like
something out of the aquarium.
351
00:23:38,040 --> 00:23:41,940
And in the center of the tank is
a rod, and seeds of quartz
352
00:23:41,940 --> 00:23:42,980
are placed on the rod.
353
00:23:42,980 --> 00:23:44,880
And the thing very
slowly rotates
354
00:23:44,880 --> 00:23:46,510
around in this solution.
355
00:23:46,510 --> 00:23:49,770
And on the rod, eventually, are
single crystals of quartz
356
00:23:49,770 --> 00:23:50,620
that are this size.
357
00:23:50,620 --> 00:23:54,290
And its a very spectacular
thing to see.
358
00:23:54,290 --> 00:23:58,270
In any case, symmetry
3 2, and the moduli
359
00:23:58,270 --> 00:24:01,370
that are 0 and non-zero.
360
00:24:01,370 --> 00:24:12,000
If we refer the reference axes
to a set of coordinates with
361
00:24:12,000 --> 00:24:15,150
x1 in this direction, and x2--
362
00:24:15,150 --> 00:24:17,960
since it has to be orthogonal
to x1-- in between the
363
00:24:17,960 --> 00:24:21,360
two-fold axes, and x3.
364
00:24:21,360 --> 00:24:30,130
And for all materials of
commerce that are anisotropic,
365
00:24:30,130 --> 00:24:33,070
there has to be some standard
for defining
366
00:24:33,070 --> 00:24:34,330
the choice of axes.
367
00:24:34,330 --> 00:24:38,820
For example, crystallographers
would say the unique axis
368
00:24:38,820 --> 00:24:42,090
should be along the z direction,
the x3 direction.
369
00:24:42,090 --> 00:24:46,490
But why isn't x1 and x2 in
between the two-fold axis?
370
00:24:46,490 --> 00:24:50,640
There's some professional
society that is responsible
371
00:24:50,640 --> 00:24:54,500
for giving standards for
representing property
372
00:24:54,500 --> 00:24:56,930
measurements in some mutually
agreed upon form.
373
00:24:56,930 --> 00:25:00,550
And this is the standard set
of axes for the quartz and
374
00:25:00,550 --> 00:25:02,200
symmetry 3, 2.
375
00:25:02,200 --> 00:25:10,610
The moduli, dij, not dijk, but
dij, these two are constrained
376
00:25:10,610 --> 00:25:12,940
to be equal.
377
00:25:12,940 --> 00:25:13,860
This one is 0.
378
00:25:13,860 --> 00:25:28,509
This is minus 0.67, 0, 0, 0, 0,
0, 0, 0.67, 4.6, 0, 0, 0,
379
00:25:28,509 --> 00:25:30,994
0, 0, 0, 0.
380
00:25:30,994 --> 00:25:35,340
One of the strange materials
for which no field
381
00:25:35,340 --> 00:25:41,450
can create a strain--
382
00:25:41,450 --> 00:25:43,810
field along x3 cannot
create a strain.
383
00:25:43,810 --> 00:25:48,570
And these are all in units of
10 to the minus 12 coulombs
384
00:25:48,570 --> 00:25:49,890
per Newton.
385
00:25:53,330 --> 00:25:57,860
An example they give you here,
if you apply a field of 100
386
00:25:57,860 --> 00:26:01,110
volts per centimeter, which is
not terribly large, but would
387
00:26:01,110 --> 00:26:02,950
be comparable to what
you have in some
388
00:26:02,950 --> 00:26:05,140
electronic device, perhaps.
389
00:26:05,140 --> 00:26:12,040
So this, since our units are
MKS, this would correspond to
390
00:26:12,040 --> 00:26:14,370
10 to the 4 volts per meter.
391
00:26:17,780 --> 00:26:23,810
The strain epsilon 1, which
is d11 times e1.
392
00:26:23,810 --> 00:26:28,510
It turns out to be minus 2.3,
which means it contracts.
393
00:26:28,510 --> 00:26:35,580
That's the significance of the
negative times 10 to the 4.
394
00:26:35,580 --> 00:26:38,310
And that turns out to be
10 to the minus 12
395
00:26:38,310 --> 00:26:39,560
times 10 to the fourth.
396
00:26:42,910 --> 00:26:49,900
That turns out to be a strain
of minus 2.3 times 10 to the
397
00:26:49,900 --> 00:26:52,900
minus eighth.
398
00:26:52,900 --> 00:26:57,800
10 to the minus eighth is not
exactly a large point strain.
399
00:26:57,800 --> 00:27:01,550
You're not going to see the
crystal wafer twitch and jump,
400
00:27:01,550 --> 00:27:04,210
if you apply a field
of 100 volts on it.
401
00:27:04,210 --> 00:27:09,450
But yet, even a strain of
this sort is more than
402
00:27:09,450 --> 00:27:12,660
enough to be useful.
403
00:27:12,660 --> 00:27:19,540
But this is just to illustrate
that quartz is not the most
404
00:27:19,540 --> 00:27:21,495
sensitive of piezoelectric
materials.
405
00:27:24,750 --> 00:27:31,660
Another one that I give you
data for is so-called ADP.
406
00:27:31,660 --> 00:27:35,360
And this is a widely
used material.
407
00:27:35,360 --> 00:27:37,665
This is ammonium dihydrogen
phosphate.
408
00:27:51,080 --> 00:27:54,780
And this is one of the family of
salts that have very large
409
00:27:54,780 --> 00:27:57,420
piezoelectric responses.
410
00:27:57,420 --> 00:28:01,180
The nice part about it is that
it's water soluble, so you can
411
00:28:01,180 --> 00:28:05,640
grow very, very large crystals
easily from solution.
412
00:28:05,640 --> 00:28:10,040
The nasty part about it is that
it is water soluble, so
413
00:28:10,040 --> 00:28:13,230
you have to be careful to
protect this material from
414
00:28:13,230 --> 00:28:15,990
moisture if you're going to use
it in any sort of device.
415
00:28:15,990 --> 00:28:21,180
But if we look at the moduli,
relative to the standard axes,
416
00:28:21,180 --> 00:28:25,720
and this has point
group 4-bar 2m.
417
00:28:25,720 --> 00:28:30,655
So x3 is taken along
the 4-bar access.
418
00:28:30,655 --> 00:28:35,010
Then there are two-fold axes
and orientations like this.
419
00:28:35,010 --> 00:28:41,280
And since they are orthogonal,
you can take both x1 and x2
420
00:28:41,280 --> 00:28:43,260
along the two-fold axes.
421
00:28:43,260 --> 00:28:54,264
And the numbers here for dij,
are 0, 0, 0, 1.7, 0, 0, 0, 0,
422
00:28:54,264 --> 00:28:57,350
0, 0, 1.7, 0.
423
00:28:57,350 --> 00:29:01,140
This is one of the interesting
tensors where there's a
424
00:29:01,140 --> 00:29:04,930
diagonal row of non-zero terms
off on the right hand side.
425
00:29:04,930 --> 00:29:07,170
And finally, the big surprise
is the third
426
00:29:07,170 --> 00:29:10,540
modulus this is 51.7.
427
00:29:10,540 --> 00:29:12,690
So you can see this has
a very strong effect.
428
00:29:12,690 --> 00:29:17,070
This is over 10 times
the maximum
429
00:29:17,070 --> 00:29:19,270
piezoelectric modulus in quartz.
430
00:29:19,270 --> 00:29:23,040
So this is a material that's
very commonly used in
431
00:29:23,040 --> 00:29:24,340
transducers.
432
00:29:24,340 --> 00:29:27,704
This is again times ten to the
minus 12 coulombs per Newton.
433
00:29:32,140 --> 00:29:36,480
One of the very, very exciting
developments in recent years
434
00:29:36,480 --> 00:29:43,000
is a class of materials
that are perovskites.
435
00:29:43,000 --> 00:29:46,435
And they are very, very new.
436
00:29:46,435 --> 00:29:48,710
I gave you the reference to
the first one that was
437
00:29:48,710 --> 00:29:53,970
reported, and that was just
in the spring of 2000.
438
00:29:53,970 --> 00:29:55,540
And these are perovskites.
439
00:29:55,540 --> 00:30:03,090
Perovskites are materials that
in the type form are cubic.
440
00:30:03,090 --> 00:30:06,840
But depending on composition
and temperature, they can
441
00:30:06,840 --> 00:30:11,530
transform to a distorted version
of this very simple
442
00:30:11,530 --> 00:30:13,890
cubic structure that
is tetragonal.
443
00:30:13,890 --> 00:30:17,480
And this material can exhibit
piezoelectric effects.
444
00:30:17,480 --> 00:30:21,050
And whether it distorts or not
depends on the relative sizes
445
00:30:21,050 --> 00:30:22,840
of what goes into
the perovskite.
446
00:30:22,840 --> 00:30:28,800
Perovskite has a composition
ABO3 like barium titanate is
447
00:30:28,800 --> 00:30:30,050
one example.
448
00:30:32,270 --> 00:30:36,410
And the material has two
different cations.
449
00:30:36,410 --> 00:30:39,750
And they have different
valences so they have
450
00:30:39,750 --> 00:30:41,560
different sizes.
451
00:30:41,560 --> 00:30:46,720
And I won't bother to describe
the structure, but it is only
452
00:30:46,720 --> 00:30:56,940
a very restricted locus in the
field RA versus RB, where both
453
00:30:56,940 --> 00:31:00,670
the A and the B can remain in
contact with the oxygen
454
00:31:00,670 --> 00:31:01,445
without distortion.
455
00:31:01,445 --> 00:31:04,680
And it turns out to be a line
that does something like that.
456
00:31:04,680 --> 00:31:07,250
Any other perovskite-- and there
are lots of them in this
457
00:31:07,250 --> 00:31:09,260
field of radii--
458
00:31:09,260 --> 00:31:11,990
has to have one of the ion
sort of flopping around.
459
00:31:11,990 --> 00:31:15,390
And if that gets too serious,
the structure distorts so that
460
00:31:15,390 --> 00:31:19,330
all these ions can remain in
contact with the oxygen.
461
00:31:19,330 --> 00:31:22,130
OK in order to do that,
many of them distort
462
00:31:22,130 --> 00:31:23,710
to tetragonal forms.
463
00:31:23,710 --> 00:31:27,840
Others distort to super
structures, which have very,
464
00:31:27,840 --> 00:31:29,940
very large unit cells.
465
00:31:29,940 --> 00:31:35,880
But in any case, when you're
right at the phase boundary
466
00:31:35,880 --> 00:31:39,460
between the distorted structure
and the true
467
00:31:39,460 --> 00:31:43,340
perovskite structure, these
materials sometimes have very,
468
00:31:43,340 --> 00:31:47,220
very complicated x solutions of
the two phases, on a very
469
00:31:47,220 --> 00:31:48,850
sort of fine scale.
470
00:31:48,850 --> 00:31:53,480
And it's not known exactly why
they have this property.
471
00:31:53,480 --> 00:31:59,000
But they have absolutely
enormous piezoelectric moduli,
472
00:31:59,000 --> 00:32:01,620
very close to this
phase boundary.
473
00:32:01,620 --> 00:32:06,590
And the references that I give
you-- here-- is a compound
474
00:32:06,590 --> 00:32:11,540
that is a lead titanium
zinc niobate.
475
00:32:15,450 --> 00:32:18,610
And the complicated composition
is to get you
476
00:32:18,610 --> 00:32:20,340
close to this phase boundary.
477
00:32:20,340 --> 00:32:29,380
This has a d333 that is greater
than 2,000 picocuries
478
00:32:29,380 --> 00:32:30,650
per Newton.
479
00:32:34,590 --> 00:32:39,960
And pico is 10 to
the minus 12.
480
00:32:39,960 --> 00:32:48,510
So this is a piezoelectric
electric modulus that is 10 to
481
00:32:48,510 --> 00:32:52,435
the 3 times d11 for quartz.
482
00:32:58,490 --> 00:33:01,430
So 3 orders of magnitude
stronger than this very
483
00:33:01,430 --> 00:33:03,340
commonly used piezoelectric
material.
484
00:33:05,860 --> 00:33:12,390
Some of these materials have
strains getting close to 1%.
485
00:33:12,390 --> 00:33:15,630
So this is something that will
actually twitch on the lab
486
00:33:15,630 --> 00:33:18,900
bench, when you apply
a field to it.
487
00:33:18,900 --> 00:33:20,450
So these are entirely new.
488
00:33:20,450 --> 00:33:24,040
People still don't know the
origin of this behavior.
489
00:33:24,040 --> 00:33:28,720
And it's still under
considerable study.
490
00:33:28,720 --> 00:33:31,120
So this is a new family
of materials.
491
00:33:31,120 --> 00:33:34,850
It's very exciting, and
undergoing a lot of
492
00:33:34,850 --> 00:33:38,940
investigation and development
work at the moment.
493
00:33:38,940 --> 00:33:41,100
Alright, we are almost
out of time.
494
00:33:41,100 --> 00:33:42,910
Time goes fast when
you're having fun.
495
00:33:46,730 --> 00:33:50,740
Let me raise the question that
we'll consider next time,
496
00:33:50,740 --> 00:33:52,780
which will be one of
our last lectures.
497
00:33:52,780 --> 00:34:00,000
And that is, is it possible to
create representation surfaces
498
00:34:00,000 --> 00:34:03,300
that tell you how the
piezoelectric properties of a
499
00:34:03,300 --> 00:34:07,300
particular material will
vary with direction?
500
00:34:07,300 --> 00:34:09,219
Well, vary with what?
501
00:34:09,219 --> 00:34:13,110
Well, we talk about the direct
piezoelectric effect.
502
00:34:13,110 --> 00:34:18,770
This gives us components of--
503
00:34:18,770 --> 00:34:23,489
well let's look at the
simpler one, in
504
00:34:23,489 --> 00:34:25,449
terms of what we apply.
505
00:34:25,449 --> 00:34:28,929
We have the converse
piezoelectric effect that says
506
00:34:28,929 --> 00:34:36,290
that epsilon ijk is going to
be dijk times e sub i.
507
00:34:39,179 --> 00:34:43,710
So we've got a piece of
material, and we apply a
508
00:34:43,710 --> 00:34:46,915
field, e sub i.
509
00:34:46,915 --> 00:34:50,780
So we can vary this in space
relative to a coordinate
510
00:34:50,780 --> 00:34:54,646
system x1, x2, x3.
511
00:34:54,646 --> 00:34:57,575
But how in the world are we
going to show what happens?
512
00:35:00,800 --> 00:35:02,720
So I always do an extra
thing in here.
513
00:35:13,730 --> 00:35:18,540
There are 9 components
to the strain
514
00:35:18,540 --> 00:35:22,370
tensor, which is symmetric.
515
00:35:22,370 --> 00:35:24,865
So they're really six responses
that are unique.
516
00:35:27,890 --> 00:35:30,210
So yes, we can define
the direction
517
00:35:30,210 --> 00:35:33,600
of the applied field.
518
00:35:33,600 --> 00:35:37,070
But they're going to be
6 different strains.
519
00:35:37,070 --> 00:35:40,910
So we're going to need six
representation surfaces.
520
00:35:40,910 --> 00:35:45,320
One for each of the three
tensile strains, and one for
521
00:35:45,320 --> 00:35:48,510
each of the three
shear strains.
522
00:35:48,510 --> 00:35:51,060
So you can't do it with
a single surface.
523
00:35:51,060 --> 00:35:58,050
So you can't do much other than
say, there are certain
524
00:35:58,050 --> 00:36:05,220
responses which are intended
to emphasize one particular
525
00:36:05,220 --> 00:36:12,400
sort of strain or one
particular sort of
526
00:36:12,400 --> 00:36:13,040
polarization.
527
00:36:13,040 --> 00:36:18,060
So one of the things we might do
is to cut a very thin plate
528
00:36:18,060 --> 00:36:20,700
of something like quartz,
and subject it
529
00:36:20,700 --> 00:36:25,010
to a uniaxial stress.
530
00:36:25,010 --> 00:36:28,050
So let's say sigma along
the x3 axis.
531
00:36:28,050 --> 00:36:31,860
So we're looking at a very
restricted strain tensor.
532
00:36:31,860 --> 00:36:36,530
That's 0, 0, 0, 0, 0,
0, 0, 0, sigma 3.
533
00:36:39,500 --> 00:36:41,710
And in response to that strain,
there are going to be
534
00:36:41,710 --> 00:36:46,320
three different components
of the polarization.
535
00:36:46,320 --> 00:36:51,740
Polarization is manifested as
a charge per unit area.
536
00:36:51,740 --> 00:36:54,310
So if we make a very
thin plate--
537
00:36:54,310 --> 00:36:56,610
to be sure there will
be charges induced
538
00:36:56,610 --> 00:37:00,580
on these thin edges--
539
00:37:00,580 --> 00:37:04,790
but if it's got a surface area
that's a large compared to the
540
00:37:04,790 --> 00:37:08,110
area of these thin edges, we
are going to be measuring
541
00:37:08,110 --> 00:37:10,200
primarily p3--
542
00:37:10,200 --> 00:37:13,160
the component of polarization
that's
543
00:37:13,160 --> 00:37:14,970
normal to this surface--
544
00:37:14,970 --> 00:37:20,070
and that might have a charge
per unit area that is
545
00:37:20,070 --> 00:37:22,880
comparable to these other
two charged surfaces.
546
00:37:22,880 --> 00:37:28,120
But because the area by design
of our specimen is so large,
547
00:37:28,120 --> 00:37:31,650
the response that would be most
easy to detect, and which
548
00:37:31,650 --> 00:37:38,280
would be the largest response,
by design, would be p3, which
549
00:37:38,280 --> 00:37:42,620
is the charge per unit
area on this surface.
550
00:37:42,620 --> 00:37:45,420
So this is an effect we can
define for a particular
551
00:37:45,420 --> 00:37:51,370
sample, and for a particular
special form of the
552
00:37:51,370 --> 00:37:53,290
generalized force.
553
00:37:53,290 --> 00:37:59,940
And we then can ask, what is
the value of the single
554
00:37:59,940 --> 00:38:04,180
modulus that relates
p3 to sigma 3?
555
00:38:04,180 --> 00:38:05,970
And that's a question
we can ask.
556
00:38:05,970 --> 00:38:14,070
And we can plot that response as
a function of direction of
557
00:38:14,070 --> 00:38:19,410
a plate that we consider as
being cut out of a single
558
00:38:19,410 --> 00:38:23,080
crystal, and different
orientations, and then ask how
559
00:38:23,080 --> 00:38:24,030
this modulus--
560
00:38:24,030 --> 00:38:25,930
which connects the two--
561
00:38:25,930 --> 00:38:29,900
changes with the orientation
of x3.
562
00:38:29,900 --> 00:38:33,050
So that is a question
we can ask.
563
00:38:33,050 --> 00:38:35,940
And these surfaces are
absolutely wild, highly
564
00:38:35,940 --> 00:38:39,130
anisotropic, can be identically
zero in certain
565
00:38:39,130 --> 00:38:42,600
special directions, and they
are very interesting, and a
566
00:38:42,600 --> 00:38:43,940
lot of fun to look at.
567
00:38:43,940 --> 00:38:46,580
So we'll take a quick look at a
couple of those, which won't
568
00:38:46,580 --> 00:38:49,690
come as a surprise because
they're already worked out for
569
00:38:49,690 --> 00:38:50,940
you in the notes.
570
00:38:53,160 --> 00:38:55,520
So we'll take a look
at one of those.
571
00:38:55,520 --> 00:38:58,640
And in the problem set, I invite
you to amuse yourself
572
00:38:58,640 --> 00:39:02,700
by looking at such
representation surfaces for
573
00:39:02,700 --> 00:39:04,920
two other point groups.
574
00:39:04,920 --> 00:39:08,550
And with that, having kept
you til five after
575
00:39:08,550 --> 00:39:10,140
the hour I will quit.